The long root geometry A n, 1, n (𝕂) for the special linear group SL (n + 1, 𝕂) admits an embedding in the (projective space of) the vector space of the traceless square matrices of order n + 1 with entries in the field 𝕂, usually regarded as the natural embedding of A n, 1, n (𝕂). S. Smith and H. Völklein in 10 have proved that the natural embedding of A 2, 1, 2 (𝕂) is relatively universal if and only if 𝕂 is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of A 2, 1, 2 (𝕂) which covers the natural one, but that information is not sufficient to exhaustively describe it. The “if” part of Smith-Völklein’s result also holds true for any n, as proved by Völklein in 13 in his investigation of the adjoint modules of Chevalley groups. In this paper we give an explicit description of the relatively universal embedding of A n, 1, n (𝕂) which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to 𝔡 + n 2 + 2 n where 𝔡 is the transcendence degree of 𝕂 over its minimal subfield (if char (𝕂) = 0) or the generating rank of 𝕂 over 𝕂 p (if char (𝕂) = p > 0). Accordingly, both the “if” and the “only if” part of Smith-Völklein’s result hold true for every n ≥ 2
Cardinali et al. (Tue,) studied this question.